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22
RISK ASSESSMENT AND OPTIMIZATION OF ROAD TUNNELS
Milan Holicky •
Klokner Institute, CTU in Prague, Solínova 7, 166 08 Praha 6, Czech Republic, Tel: +420 224 310 208, Fax: +420 224 355 232, [email protected]
Abstract
Probabilistic methods of risk optimization are applied to specify the most effective arrangements of road tunnels. The total consequences of alternative arrangements are assessed using Bayesian networks supplemented by decision and utility nodes. It appears that the optimization may provide valuable information for a rational decision concerning number of escape routes. Discount rate seems to affect the total consequences and the optimum arrangements of the tunnels more significantly than number of escape routes.
Key words
Risk assessment, social risks, economic consequences, road tunnels, Bayessian network, optimization, escape routes, discount rate, expected life time
INTRODUCTION
Tunnel structures usually represent complex technical systems that may be exposed to hazard situations leading to unfavourable events with serious consequences. Minimum safety requirements for tunnels in the trans-European road network are provided in the Directive of the European Parliament and of the Council 2004/54/ES [1]. The Directive also gives recommendations concerning risk management, risk assessment and analysis.
Methods of risk assessment and analysis are more and more frequently applied in various technical systems [2,3] including road tunnels [4]. This is a consequence of recent tragic events in various tunnels and of an increasing effort to take into account social, economic and ecological consequences of unfavourable events [2,3,4]. Available national and international documents [5] to [10] try to harmonise general methodical principles and terminology that can be also applied in the risk assessment of road tunnels. The submitted contribution, based on previous studies [11] to [17] and recent PIARC working documents, attempts to apply methods of probabilistic risk optimization using Bayesian networks supplemented by decision and utility nodes [18]. It appears that Bayesian networks provide an extremely effective tool for investigating the safety of road tunnels.
GENERAL PROCEDURE OF RISK ASSESSMENT
The main components of the whole risk management consist of risk assessment and risk control. The risk control is outside the scope of this paper. The risk assessment consists of risk analysis and risk evaluation. A general procedure of risk assessment is shown in Figure 1 indicating a flowchart of the main steps. The flowchart is adopted from ISO document [9] and from recent working materials of PIARC/C3.3/WG2. The contents of individual steps are mostly obvious from the relevant key words used for description of the flowchart. Two key steps of the risk analysis, probability analysis and risk estimation are shortly described below.
PROBABILITY ANALYSIS
Probabilistic methods of risk analysis are based on the concept of conditional probabilities Pfi = P{F|H.} of the event F providing a situation Hi occurs [1, 3]. In general this probability can be found using statistical data, experience or theoretical analysis of the situation Hi.
If the situation Hi occurs with the probability P(H) and the event F during the situation Hi occurs with the probability P(F|Hi), then the total probability PF of the event F is given as
Pf = £ P(F | Hi )P(H.) (1)
i
Equation (1) makes it possible to harmonize partial probabilities P(F|H.) P(H.) related to the situation H..
The main disadvantage of the purely probabilistic approach is the fact that possible consequences of the events F related to the situation Hi are not considered. Equation (1) can be, however, modified to take the consequences into account.
Figure 1. Flowchart of iterative procedure for the risk assessment (adopted from [9]) RISK ESTIMATION
A given situation Hi may lead to a set of events Eij (for example fully developed fire, explosion), which may have social consequences RiJ or economic consequences Cij. It is assumed that the consequences Rij and Cij are unambiguously assigned to events Eij. If the consequences include only social components RiJ, then the total expected risk R is given as [11]
R = X R P(Eij | Hi )P(Hi ) (2)
If the consequences include only economic consequences Cij, then the total expected consequences C are given as
c = S CjP(Ej | Hi )P(Hi ) (3)
If criteria Rd and Cd are specified, then acceptable total consequences should satisfy the conditions
R < Rd and C < Cd (4)
that supplement the traditional probabilistic condition Pf < Pfd.
When the criteria are not satisfied, then it may be possible to apply a procedure of risk treatment as indicated in Figure 1. For example additional escape routes may be provided. Such measures might, however, require considerable costs, which should be considered when deciding about the optimum measures.
PRINCIPLES OF RISK OPTIMIZATION
The total consequences Ctot(k,p,n) relevant to the construction and performance of the tunnel are generally expressed as a function of the decisive parameter k (for example of the number k of escape routes), discount rate p (commonly aboutp ~ 0,03) and life time n (commonly n = 100 let). The decisive parameter k usually represents a one-dimensional or multidimensional quantity significantly affecting tunnel safety.
The fundamental model of the total consequences may be written as a sum of partial consequences as
Ctot(k,p,n) = R(k,p,n) + C0 +^C(k) (5)
In equation (5) R(k,p,n) denotes expected social risk that is dependent on the parameter k, discount rate p and life time n. C0 denotes the basic of construction cost independent of k, and AC(k) additional expenses dependent on k. Equation (5) represents, however, only a simplified model that does not reflect all possible expenses including economic consequences of different unfavourable events and maintenance costs.
The social risk R(k,p,n) may be estimated using the following formulae
R(k,p,n) = N(k) Zi Q(p,n), Q(p,n) = 1 ~ 1(' + p\ (6)
1 -1(1 + p)
In equation (6) N(k) denotes number of expected fatalities per one year (dependent on k), Z1 denotes acceptable expenses for averting one fatality, and p the discount rate (commonly within the interval from 0 to 5 %). The quotient q of the geometric row is given by the fraction q = 1/(1+p). The discount coefficient Q(p,n) makes it possible to express the actual expenses Z1 during a considered life time n in current cost considered in (5). In other words, expenses Z1 in a year i correspond to the current cost Z1 q1. The sum of the expenses during n years is given by the coefficient Q(p,n).
A necessary condition for the minimum of the total consequences (5) is given by the vanishing of the first derivative with respect to k that may be written as
^ZQXp.n) = -*§» (7)
dk dk
In some cases this condition may not lead to a practical solution, in particular when the discount rate p is small (a corresponding discount coefficient Q(p,n) is large) and there is a limited number of escape routes k that can not be arbitrary increased.
STANDARDIZED CONSEQUENCES
The total consequences given by equation (5) may be in some cases simplified to a dimensionless standardized form and the whole procedure of optimization may be generalized. Consider as an example the optimization of the number k of escape routes. It is assumed that involved additional costs AC(k) due to k may be expressed as the product k C1, where C1 denotes cost of one escape route. If C1 is approximately equal to expenses Z1 (assumed also in [14]), equation (5) may be written as
Ctot(k,p,n) = N(k) C1 Q(p,n)+ C0 + k C1 (8)
This function can be standardized as follows
K(k, p, n) = Ctot (k, A n) - C° = N(k)Q(p, n) + k (9)
C1
Obviously both variables Ctot(k,p,n) and K(k,p,n) are mutually uniquely dependent and have the extremes (if exist) for the same number of escape routes k. A necessary condition for the extremes follows from (7) as
dN (k ) dk
1
1 -1(1 + p)
Q( p, n) 1 -1(1 + p)n
(10)
An advantage of standardized consequences is the fact that it is independent of C0 and C1. It is only assumed that C1 ~ Zj is a time invariant unit of the total consequences.
MODEL OF A TUNNEL
A road tunnel considered here (Figure 2) is partly adopted from a recent study [14]. It is assumed that the tunnel has the length of 4000 m and two traffic lanes in one direction are used by heavy goods vehicles HGV, dangers goods vehicles DGV and Cars.
The main model
Input Informtion
Number of collisions per year and the whole tunnel (4 km)
Decision concerning traffic pattern
Sub-models for HGV. DGV and Cars
Risks due to fatalities
Sum of risks and costs
The totalcos^)
Figure 2. Main model of the tunnel
The total traffic intensity in one direction is 20x106 vehicles per year (27 400 vehicles in one lane per day). The number of individual types of vehicles is assumed to be HGV:DGV:Cars = 0,15:0,01:0,84. The frequency of series accidents for basic traffic conditions (that might be possibly improved) is considered as 1 x10-7 per one vehicle and one km [14], thus 8 accidents in the tunnel per year.
The main model of the tunnel shown in Figure 2 includes three sub-models for HGV, DGV and Cars, which describe individual hazard scenarios. The Bayesian networks used here need a number of other input data. Some of them are adopted from the study [14] (based on event tree diagram), the other are estimated or specified using expert judgement. Detailed description of the model is outside the scope of this contribution.
RISK OPTIMIZATION
Risk optimization of the above described tunnel is indicated for selected input data in Figure 3, Figure 4 and 5. Figure 3 shows variation of the components of standardized total consequences Kk,p,n) with number of escape routes k for a common value of the discount rate p = 0,03 and assumed life time n = 100 years.
Figure 3. Variation of the components of standardized total consequences K(k,p,n) with k for the discount
rate p = 0,03 and life time n = 100 years
Figure 4 shows variation of the standardized total consequences K(k,p,n) with k for selected discount rate p life time n = 50 years only, Figure 5 shows similar curves as Figure 4 but for expected life time n = 100 years (common value). Both Figures 4 and 5 clearly indicate that the discount rate p and life time n affect the total consequences more significantly than the number of escape routes k. It appears that the total consequences considerably increase with increasing n. For small discount rates p < 0.01 and life time n = 100 years the total consequences decrease monotonously with increasing k and for k < 39 (the distance of escape routes up to 100 m) do not reach its minimum. Therefore, in this case condition (10) does not lead to a practical solution.
Standardized consequences
120 110 100 90 80 70 60 50 40
I iscount rate
t \ p = 0,00
X \ X X p = 0,01 ____
W - « -...... p = 0,02
p = 0,03 p = 0,04 —^ _ - *-v, ^ ^^^...... m ^^^^^^
p = 0,05
10 20 30
Number of escape routes k
40
Figure 4. Variation of the standardized total consequences K(k,p,n) with k for selected discount rate p
life time n = 50 years
Standardized consequences
120 110 100 90 80 70 60 50 40
\
\ \ \ I iscount rate
\ \ \ V p = 0,01
X X X
-„ p = 0,02 — ^ « ^ — ~
* N n n HI 0
........ p 0,03 p = 0,04 __
p = 0,05
0
40
10 20 30
Number of escape routes k
Figure 5. Variation of the standardized total consequences K(k,p,n) with k for selected discount rate p life
time n = 100 years
0
Figure 6 shows variation of the total consequences K(k,p,n) with number of escape routes k and discount rate p assuming again expected life n = 100 years.
Figure 6. Variation of the standardized total consequences K(k,p,n) with k for selected discount rate p life
time n = 100 years
Figure 6 clearly illustrates previous finding that the discount rate p affects the total consequences K(k,p,n) more significantly than the number of escape routes k.
CONCLUSIONS
Similarly as in case of other technical systems the risk assessment of road tunnels commonly includes
- definition of the system
- hazard identification
- probability and consequences analysis
- risk evaluation and possible risk treatment
Two kinds of criteria commonly applied in the risk assessment of road tunnels relate to:
- expected individual risk
- cumulative social risk (fN curves)
Probabilistic risk optimization based on the comparison of social and economic consequences may provide background information valuable for a rational decision concerning effective safety measures of road tunnels. It appears that the discount rate and assumed life time may affect the total consequences and the optimum arrangements of the tunnels more significantly than the number of escape routes. However, further investigations of relevant input data concerning social and economic consequences are needed.
Standardized consequences
ACKNOWLEDGEMENT
This study is a part of the project Isprofond 5006210025 „SAFETY OF ROAD TUNNELS IN
ACCORDANCE WITH DIRECTIVES OF EU" supported by National fond of traffic infrastructures.
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